Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [calculus]

For basic questions about limits, continuity, derivatives, differentiation, integrals, and their applications, mainly of one-variable functions.

Calculus is the branch of mathematics studying the rate of change of quantities, which can be interpreted as slopes of curves, and the lengths, areas and volumes of objects.

Calculus is divided into differential and integral calculus, which are concerned with derivatives

$$\frac{\mathrm{d}y}{\mathrm{d}x}= \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}$$

and integrals

$$\int_a^b f(x)\,\mathrm{d}x = \lim_{\Delta x \to 0} \sum_{k=0}^n f(x_k)\ \Delta x_k,$$

respectively.

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Fundamental Theorem of calculus question from Apostol

I am having some trouble with this question out of Apostol's 'Calculus' Volume 1, Section 5.5 Question 19 (Page 209). This is not a homework question, I am doing this for personal study. The question is as follows: Given a function $g$, continuous…
emjay
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Trigonometric Substitution

I have been working on some homework problems and there is one problem that has me completely stumped. The question reads: "Evaluate the integral $\int \sqrt{24+4x^2}\,dx$ using trigonometric substitution." I am really unsure how to approach this…
user124539
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Help me understand notation

Note: The inverse of $f(x)$, $f^{-1}(f(x)) = x$. My interpretation of this: $1$ is equal to the derivative of $x$ which is equal to the derivative of the inverse function. Then this next part is where it gets foggy for me... The derivative of the…
user3200098
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Differentiable only on interior points

I have been pondering this question for quite a while and I would really like to get some closure on it. Whenever I have read about differentiation of functions defined on an interval, they almost always require differentiability only being valid…
user130591
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When does product of derivatives equals derivative of products?

In general, $\frac{d}{dx}(f(x) \cdot g(x)) \neq \frac{d}{dx}f(x) \cdot \frac{d}{dx}g(x)$ When does this result hold true? My first try is to use product rule on left side and compare the two sides, but this hasn't helped at all. Any suggestions?
jj77646_piyy
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Derivative of $a^x$ from first principles

I've been trying to think for the past few days how one could differentiate $a^x$ based on the definition that $a^n$ is repeated multiplication, $a^{n/m}=(\sqrt[m]a)^n$, and $a^x$ is the completion of the above function by continuity. With a bit of…
Jack M
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evaluation of $\int\frac{1}{\sin^3 x-\cos^3 x}dx$

Evaluation of $\displaystyle \int\frac{1}{\sin^3 x-\cos^3 x}dx$ $\bf{My\; Try::}$ Given $\displaystyle \int\frac{1}{\sin^3 x-\cos^3 x}dx = \int\frac{1}{(\sin x-\cos x)\cdot (\sin^2 x-\sin x\cos x+\cos^2 x)}dx$ $\displaystyle = 2\int\frac{(\sin…
juantheron
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The chain rule, how to interpret it

I can't understand what the notation means, it doesn't make sense to me, it is too complex for me to think of. Anyways I just can't understand what $f'\bigl(g(x)\bigr)g'(x)$ means, or the crazy $dy/dx = (dy/du)(du/dx)$. I made sense of it in class…
user138246
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Chain Rule for more than two functions, general formula.

For a composition of $k$ functions, $F(x) = f_1 \circ f_2 \circ \dots \circ f_k(x)$, (I'm not sure if that is the correct notation for function composition of more than two functions), is there a general formula for finding $F'(x)$?
user124862
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Differentiating the polynomial $x^3 - 4x +6$

I am suppose to differentiate $$x^3 -4x +6,$$ but I can not get the correct answer. I get $\frac{\mathrm{d} }{\mathrm{d} x}(x^3) = 3x^2$ using the power rule and then $\frac{\mathrm{d} }{\mathrm{d} x}(4x) = 4$ and $\frac{\mathrm{d} }{\mathrm{d}…
user138246
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Calculation of $f(x) = \int_{0}^{\frac{\pi}{4}}\ln \left(1+x\cdot \tan z\right)dz$

If $\displaystyle f(x) = \int_{0}^{\frac{\pi}{4}}\ln \left(1+x\cdot \tan z\right)dz,$ where $x>-1$. Then value of $\displaystyle f\left(\frac{1}{2}\right)+f\left(\frac{1}{3}\right) = $ $\bf{My\; Try::}$ Given $\displaystyle f(x) =…
juantheron
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Calculation of $ \int_{0}^{\sqrt{n}}\lfloor t^2 \rfloor dt\;$, where $n\in \mathbb{N}$ and $\lfloor x \rfloor = $ floor function of $x$.

Calculation of $\displaystyle \int_{0}^{\sqrt{n}}\lfloor t^2 \rfloor dt\;$, where $n\in \mathbb{N}$ and $\lfloor x \rfloor = $ floor function of $x$. $\bf{My\; Try}::$ Let $t^2 = u$ and $\displaystyle dt = \frac{du}{2\sqrt{u}}$ and changing Limits,…
juantheron
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Basic Calculus counterexample

We know that if we have two real differentiable functions $\;f,g\;$ on an interval $\;I\subset \Bbb R\;$ s.t. $\;h'(x)=g'(x)\;\;\forall\,x\in I\;$ , we then get that $\;f(x)=g(x)+C\;$ on $\;I\;$ , $\;C=$ a constant, and from here we get, under the…
DonAntonio
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If $f(0)=0 ,f(1)=1 ,f'(0)=f'(1)=0$, then $|f''(x)|>4$

Let $ f $ be a twice differentiable function with $ f (0) = 0, f (1) = 1$ and $ f '(0) = f' (1) = 0 $, then $ 4 \leq | f'' (x ) | $, for some $ x \in [0,1] $. I tried using the mean value theorem for derivatives with $ \dfrac{f '(1)-f' (0)}{f (1)-f…
kEoz
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Volume of a surface of revolution with curve in parametric form

On Wikipedia, I recently stumbled upon a method of obtaining the volume of a solid of revolution generated by a curve in parametric form, which was useful in my case because I had a curve I had trouble representing as an equation of 2 variables. …
Mike
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